We give elementary proofs for the Apagodu-Zeilberger-Stanton-Amdeberhan-Tauraso congruences r Xp1 n=0 2n n p Xr1 n=0 2n n mod p2, r Xp1 n=0 sp X1 m=0 n + m m 2 p Xr1 m=0 Xs1 n=0 n + m m 2 mod p2, where p is an odd prime, r and s are nonnegative integers, and p = 8 >< >: 0, if p 0 mod 3; 1, if p 1 mod 3; 1, if p 2 mod 3.
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机译：我们为apagodu-zeilberger-stanton-amdeberhan-tauraso同时提供基本证据R XP1 n = 0 2N NP XR1 n = 0 2N N MOM P2，R XP1 n = 0 SP X1 m = 0 n + mm 2 p xr1 m = 0 xs1 n = 0 n + mm 2 mod p2，其中p是奇数素数，r和s是非负整数，并且p = 8> <>：0，如果p 0 mod 3; 1，如果p 1 mod 3; 1，如果p 2 mod 3。
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